A Note on the Fundamental Theorem of Algebra

Abstract

The algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First, every polynomial with odd degree and real coefficients has a real root. Second, every nonnegative real number has a square root. Shipman [‘Improving the fundamental theorem of algebra’, Math. Intelligencer29(4) (2007), 9–14] showed that the assumption about odd degree polynomials is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed. In this paper, we give a simpler proof of this result of Shipman.

Discussion on the Structure of the Reduced Residue Class Additive Group in Cyclic Group

“…Suppose henceforth that 1 µ > . Then the order p µ − and p-1 yields, according to the theorem in Mohsen Aliabadi's study "A Note on the Fundamental Theorem of Algebra" [2], an automorphism of each direct factor of Μ and ' µ Β respectively, and consequently of these groups themselves. Hence, we have the main conclusions of this paper:…”

“…The additive group [1][2][3] of the residue class ring [4][5] mod m is, up to isomorphism, the cyclic group of order m generate, say, by the residue class 1 mod m. For finite group, the direct sum decomposition [6] with respect to the prime power appearing in m is simply a special case of the basis theorem for finite abelian groups, according to the sum of cyclic subgroups of prime power order. Here, however, the direct sum of each cyclic subgroup belongs to the same prime number, rather than the direct sums of the cyclic subgroups themselves, which are uniquely determined.…”

Strictly Real Fundamental Theorem of Algebra Using Polynomial Interlacing

“…Even though the minimal algebraic elements for proving the FTA have been elucidated [1,4,8], the minimal analysis platform to prove the FTA has not been characterised. We will show that the basic properties of the real number field suffice to prove Theorem 1.1 by considering the linear residue of the division of N n=0 c n x n by (x 2 − ax − b) for a, b ∈ R:…”